Properties

Label 1568.2.a.x
Level $1568$
Weight $2$
Character orbit 1568.a
Self dual yes
Analytic conductor $12.521$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \(x^{4} - 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} -2 \beta_{1} q^{5} + 7 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} -2 \beta_{1} q^{5} + 7 q^{9} -\beta_{3} q^{11} + 2 \beta_{3} q^{15} -3 \beta_{1} q^{17} -\beta_{2} q^{19} -2 \beta_{3} q^{23} + 3 q^{25} -4 \beta_{2} q^{27} -6 q^{29} -2 \beta_{2} q^{31} + 10 \beta_{1} q^{33} + 2 q^{37} -\beta_{1} q^{41} + \beta_{3} q^{43} -14 \beta_{1} q^{45} + 2 \beta_{2} q^{47} + 3 \beta_{3} q^{51} + 6 q^{53} + 4 \beta_{2} q^{55} + 10 q^{57} -\beta_{2} q^{59} + 6 \beta_{1} q^{61} + 20 \beta_{1} q^{69} -2 \beta_{3} q^{71} + 5 \beta_{1} q^{73} -3 \beta_{2} q^{75} -2 \beta_{3} q^{79} + 19 q^{81} + 3 \beta_{2} q^{83} + 12 q^{85} + 6 \beta_{2} q^{87} -7 \beta_{1} q^{89} + 20 q^{93} + 2 \beta_{3} q^{95} -3 \beta_{1} q^{97} -7 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 28q^{9} + O(q^{10}) \) \( 4q + 28q^{9} + 12q^{25} - 24q^{29} + 8q^{37} + 24q^{53} + 40q^{57} + 76q^{81} + 48q^{85} + 80q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 4 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 8 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} + 4 \beta_{1}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.28825
0.874032
−0.874032
−2.28825
0 −3.16228 0 −2.82843 0 0 0 7.00000 0
1.2 0 −3.16228 0 2.82843 0 0 0 7.00000 0
1.3 0 3.16228 0 −2.82843 0 0 0 7.00000 0
1.4 0 3.16228 0 2.82843 0 0 0 7.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.a.x 4
4.b odd 2 1 inner 1568.2.a.x 4
7.b odd 2 1 inner 1568.2.a.x 4
7.c even 3 2 1568.2.i.y 8
7.d odd 6 2 1568.2.i.y 8
8.b even 2 1 3136.2.a.bz 4
8.d odd 2 1 3136.2.a.bz 4
28.d even 2 1 inner 1568.2.a.x 4
28.f even 6 2 1568.2.i.y 8
28.g odd 6 2 1568.2.i.y 8
56.e even 2 1 3136.2.a.bz 4
56.h odd 2 1 3136.2.a.bz 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.2.a.x 4 1.a even 1 1 trivial
1568.2.a.x 4 4.b odd 2 1 inner
1568.2.a.x 4 7.b odd 2 1 inner
1568.2.a.x 4 28.d even 2 1 inner
1568.2.i.y 8 7.c even 3 2
1568.2.i.y 8 7.d odd 6 2
1568.2.i.y 8 28.f even 6 2
1568.2.i.y 8 28.g odd 6 2
3136.2.a.bz 4 8.b even 2 1
3136.2.a.bz 4 8.d odd 2 1
3136.2.a.bz 4 56.e even 2 1
3136.2.a.bz 4 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1568))\):

\( T_{3}^{2} - 10 \)
\( T_{5}^{2} - 8 \)
\( T_{11}^{2} - 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 4 T^{2} + 9 T^{4} )^{2} \)
$5$ \( ( 1 + 2 T^{2} + 25 T^{4} )^{2} \)
$7$ 1
$11$ \( ( 1 + 2 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( ( 1 + 16 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 28 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 34 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 22 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 80 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 66 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 54 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 6 T + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 108 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 50 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 67 T^{2} )^{4} \)
$71$ \( ( 1 + 62 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + 96 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 78 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 76 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 80 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 176 T^{2} + 9409 T^{4} )^{2} \)
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